Thursday, December 27, 2007

Weak Law of Large Numbers

Let X_1,X_2,X_3,... be independent and identically distributed, each with mean \mu and variance \sigma^2. Then for any given \varepsilon \text{\textgreater }0,

P(|\bar{X}_n - \mu| \text{\textless }\: \varepsilon }) \to 1 \quad \text{as}\; n\to \infty.

Proof?

Erm... well it has a lot of nasty probability involved, are you sure you want it? I didn't actually write that law for its own sake you know! (Proof will be given later, read *VERY LATER* but if you really want it then let me know! Alternatively you could try proving the above yourself. You might want to use Markov's inequality and take g(\bar{X}_n)= (\bar{X}_n - \mu)^2 and k = \varepsilon^2.)

The real reason I wrote that law was to remind myself of something. Posts concerning negative numbers have been going around Blogistan, about this article. I have sat quietly because my own struggles with negative numbers is bizarre and still possibly exists (with respect to inequalities etc).

Yes - inequalities are indeed the reason for my post on this. I don't think I have a problem with deciding which number is bigger from -5 and -4, but today for the life of me I couldn't understand why -4 was bigger than -5! Whilst trying to understand the proof of the theorem (which I will mention in a while), the numbers -4 and -5 happened to be on my paper. As usual I talk to myself and asked myself which was bigger. Someone else happened to be listening to me at the time (a primary school kid) and so I asked her which was bigger. The kid answered -5. I queried at this choice and was told that 5 is bigger than 4. I told the child that she was incorrect, and then tried to give some silly explanation but ended up confusing myself!! Annoyed at myself I had apologised to the primary school kid before returning to the proof.

However that didn't stop me from talking to myself. Again I loudly asked, "Why is -4 bigger than -5?" (If you are wondering why the heck a doofus like me is doing a maths degree, then you are not alone in that thought!) I know that -4 is bigger but I just couldn't convince myself as to why; hence causing myself silly unwanted frustration. Finally I drew a vertical number line (as I used to in secondary school!). 5 was on top of 4 so 5 was bigger. -4 was on top of -5 therefore -4 was bigger. Yep - that was my lame explanation to myself but it knocked me out of my primary school behaviour and back to stats.

In statistics you have a lot of inequalities flying about, and it is crucial that you use the right sign. As you read above, we have Markov's inequality and another one is Chebyshev's inequality. (They give a bound for the probability - a bit like the estimation lemma in complex analysis(?)). Proofs that use these inequalities, ultimately require some manipulation of inequalities. Once again, like a couple of weeks ago, I couldn't see why the sign was the way it was (when I do post the proof you will know the exact details!)

Maybe I should have done this from day one, but today after much mindless staring, I (perversely) wrote -4 = 1 - 5. After which I wrote the inequality -5 \ge -6. Hence -4 = 1 - 5 \ge 1 -6. That might seem very easy and you are probably asking the same question about my degree again(!), but the symbol used here was the greater than sign. I had once again been sticking the less than sign everywhere, and not understanding why it had been wrong. Going back to what I did allowed me to see why I had been wrong. In future maybe I should give every inequality that I have been given, numerical values just to get the sign right!

So how many times in the past few days has the question about my degree come up? I count four - two in this post and two yesterday. Is this due to statistics or should we be really worried?! (I think it's because of probability and stats though!) Seriously these past few days have left me wondering about whether I can do any maths whatsoever! Indeed I haven't been feeling like a mathematician.

I shouldn't really be up at this time but got talking to someone about things till 1:30am. Sometimes children are the best people to talk to, i.e. people younger than you! (I hope that I am still a child yet!) I mean those who have yet to experience what you have and are fearless in their claims. They have a certain freedom in their speech and any "advice" they give you is not harmful or bitter. Not recycled and like a waiter who serves food all the time (if you follow me). I think it is natural that if I was to give "advice" i.e. share my experiences, I will obviously also warn others of what not to do. Those who have not yet got those experiences i.e. those younger than you don't hear the warnings and don't see them. I think I am talking rubbish again, but as I sat and talked with someone younger than me I realised the freedom of being young. Impossible is actually nothing. And then you grow older and realise that if you do a certain thing it will hurt you, others etc and so you lose some amount of fearlessness.

Sometimes I wonder, would I prefer being a 12 year old (nit wit!) with no real worries in the world - no responsibilities; or would I prefer being a 19 year old who seems to over think! If I was 12 and aware of the exciting maths which I have been studying then sure thing! However it is a close call to make for I definitely do not want to go back to doing ratios and what not. Sigh. It's 2:44am and I am mumbling to myself. I was meant to be revising when I was talking earlier, and exclaimed this fact to my friend. "I was meant to finish stats today!" (A common sentence from my mouth nowadays). I was then told to stop worrying and to stop thinking that I can't complete it tomorrow or that I had to complete it by tomorrow. ( I didn't remind my friend that P(completion)=0). It is nice to be understood once in a while.

The demoralizing thing for me is that as I spend time understanding stats, I still don't get it. My problem has been that I am not revising but trying to understanding what has been written. However when I try to understand material I never work linearly through my notes, and actually use a gazillion books. I have been getting carried away recently and not remaining focused on the chapter at hand. This is what causes me to spend hours at times, on trivial things and sometimes more! (Time actually flies). I really want to understand the concepts properly though - not artificially. But sadly I have ran out of time. Tomorrow I must persevere and complete 6 lectures. One lecture takes me a more than an hour to understand as well..

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Good morning! It seems that I feel asleep whilst typing that post. Well actually I thought of resting my eyes for a few minutes, and woke up to see the sun outside! After a quick read through that last paragraph I have come to a conclusion. One reason why I am really bad at statistics is because of my attitude towards the subject. Whenever it is mentioned, I don't pretend to hide my groans and pain. Honestly, I am unfortunately ruthless to this subject and my answer has always been because I can't understand it to a level of comfortableness.

The same applies to my complete miss hit with inequalities. As soon as I see a greater than sign my mind blows up and shuts down! Straight away I try to re-write it as a less than sign, then panic some and write something incorrect. Once I can get rid of this "dread" for inequalities I should be OK (or at least I hope to be). I feel silly because of this, but it is a serious problem. Inequalities are really powerful (and I have just remembered that some induction proofs make use of them). They are everywhere!! RUN FOR YOUR LIVES!

Enough drama. My attitude towards stats might never change, namely because I will never understand it and can't find anything "cool" about it. However I would love to be given a few extra days to try and understand it better. And for the record probability is MUCH better than statistics. So I was correct (for a change) in always writing "I hate stats". I hope though, that I am able to like it enough for my exam though.

And inequalities -- well if I can go for a year without making a blunder when dealing with them, I will consider myself cured!

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